The norm $N(a)$ of an $a$ element of a field extension $K/L$ is the determinant of the matrix representing multiplication by $a$. It has the following properties: $$ N(a b) = N(a)N(b) \\ N(ka)=k^n N(a) $$ where $ a,b\in L$ and $k\in K$ and $n$ is the degree of the extension.
But there is no mention of addition.
No, the field norm is not a norm in the sense of normed vector spaces.
One reason is that the field norm takes values in $L$ and vector space norms take values in $\mathbb R$.
Even when $L \subset \mathbb R$, the field norm is not a vector space norm because it can be negative.
Wikipedia offers this example: In $\mathbb Q(\sqrt{2})$, the field norm of $ 1+\sqrt{2}$ is $-1$.